5.7. Subobjects and Quotient Objects
The entire point of category theory, contrary to its name, is to unify mathematics. Mathematicians saw the same stories over and over again in algebra and topology, and one day they got sick of it and decided to start naming the patterns they were seeing. Mathematicians achieved a level of abstraction where we no longer really care about the objects, but we want to study the morphisms between them. However, in many categories, the objects are often things like groups, rings, or topological spaces; hence there are subgroups, subrings, and spaces with subset topologies which also exist inside categories we study. This presents a challenge for category theory: how do we generalize the notion of subgroups or subspaces if we always avoid explicit reference to the elements?
It turns out that the correct way to go about this is to consider the philosophy of
sub-"things": whenever
For example, in Set,
Thus we see that these monomorphisms give us sub-"things," and so we might naively say
the set of all "subobjects" of an object
However, the space of all of
these monomorphisms is huge, and also repetitive. For example, in Set, if we have
Each arrow is basically saying the same thing. How do we deal with this?
Well, we can impose an equivalence relation on this space to obtain something smaller
and more manageable.
Let
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for some monomorphism
Let
Let
If we play around with these functors long enough, we may ask the question:
What happens when, for a functor
Could we logically call
The answer is no. This is because
The diagram then commutes. But is this the only way to make it commute? Suppose with no assumption of
Then we see that
Thus we could define
However, we can recover the same concept by applying subobjects to this functor category.
In this case, we can (with laziness) say a
Unwrapping this definition, we see that a monic natural transformation in this
case is just one where each morphism
Hence we have recovered the same concept of a subfunctor in two different ones;
one in which we followed our intuition, and one in which we blinded applied the concept of a
subobject in the functor category
The previous example allows us to make the definition:
Let
is a subfunctor of
Now, perhaps unsurprisingly, the entire process above can be dualized. When we dualize, however, we obtain a generalization of the concept of quotient objects. Instead of just dualizing and being boring, we'll motivate why we'd even care for such a dual concept. \
In interesting categories such as Ab or Top, we not only have
subgroups and subspaces, but we also have quotient groups and quotient spaces.
For the case of abelian groups, we can, for any such group
For topological spaces
With these few examples, we see that it is worthwhile to generalize the concept of quotient objects; to do this however requires no explicit mention of the elements of the objects of the category. However, we can maintain the philosophy seen in the previous two examples to generalize the concept.
For an object
and call objects such objects
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Observing that
Let
A quotient object in Cat is a quotient category (from chapter 2)